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Example: Jacobi finite difference method
X(t+1)i,j=(4X(t)i,j+X(t)i-1,j+X(t)i+1,j+X(t)i,j-1+X(t)i,j+1)/(8)
for t = 0 to T-1 send X(t)(i,j) to each neighbor receive from neighbors X(t)(i-1,j), X(t)(i+1,j), X(t)(i,j-1), X(t)(i,j+1) compute X(t+1)(i,j) end
Easy parallelization, but many iterations.
Improvement: Gauss-Seidel strategy
X(t+1)i,j=(4X(t)i,j+X(t+1)i-1,j+X(t)i+1,j+X(t+1)i,j-1+X(t)i,j+1)/(8)
Elements are updated in a particular order.
(See Foster, Figures 2.4 and 2.5)